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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.ellint.ellint_intro"></a><a class="link" href="ellint_intro.html" title="Elliptic Integral Overview">Elliptic Integral Overview</a>
</h3></div></div></div>
<p>
        The main reference for the elliptic integrals is:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions
          with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards
          Applied Mathematics Series, U.S. Government Printing Office, Washington,
          D.C.
        </p></blockquote></div>
<p>
        and its recently revised version <a href="http://dlmf.nist.gov/" target="_top">NIST
        Digital Library of Mathematical Functions (DMLF)</a>, in particular
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <a href="https://dlmf.nist.gov/19" target="_top">Elliptic Integrals, B. C. Carlson</a>
        </p></blockquote></div>
<p>
        Mathworld also contain a lot of useful background information:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <a href="http://mathworld.wolfram.com/EllipticIntegral.html" target="_top">Weisstein,
          Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web Resource.</a>
        </p></blockquote></div>
<p>
        As does <a href="http://en.wikipedia.org/wiki/Elliptic_integral" target="_top">Wikipedia
        Elliptic integral</a>.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h0"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.notation"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.notation">Notation</a>
      </h5>
<p>
        All variables are real numbers unless otherwise noted.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h1"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.definition"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.definition">Definition</a>
      </h5>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint1.svg"></span>

        </p></blockquote></div>
<p>
        is called elliptic integral if <span class="emphasis"><em>R(t, s)</em></span> is a rational
        function of <span class="emphasis"><em>t</em></span> and <span class="emphasis"><em>s</em></span>, and <span class="emphasis"><em>s<sup>2</sup></em></span>
        is a cubic or quartic polynomial in <span class="emphasis"><em>t</em></span>.
      </p>
<p>
        Elliptic integrals generally cannot be expressed in terms of elementary functions.
        However, Legendre showed that all elliptic integrals can be reduced to the
        following three canonical forms:
      </p>
<p>
        Elliptic Integral of the First Kind (Legendre form)
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>

        </p></blockquote></div>
<p>
        Elliptic Integral of the Second Kind (Legendre form)
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span>

        </p></blockquote></div>
<p>
        Elliptic Integral of the Third Kind (Legendre form)
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>

        </p></blockquote></div>
<p>
        where
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint5.svg"></span>

        </p></blockquote></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
          <span class="emphasis"><em>φ</em></span> is called the amplitude.
        </p>
<p>
          <span class="emphasis"><em>k</em></span> is called the elliptic modulus or eccentricity.
        </p>
<p>
          <span class="emphasis"><em>α</em></span> is called the modular angle.
        </p>
<p>
          <span class="emphasis"><em>n</em></span> is called the characteristic.
        </p>
</td></tr>
</table></div>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top">
<p>
          Perhaps more than any other special functions the elliptic integrals are
          expressed in a variety of different ways. In particular, the final parameter
          <span class="emphasis"><em>k</em></span> (the modulus) may be expressed using a modular angle
          α, or a parameter <span class="emphasis"><em>m</em></span>. These are related by:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">k = sin  α</span>
          </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>α</span>
          </p></blockquote></div>
<p>
          So that the integral of the third kind (for example) may be expressed as
          either:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">Π(n, φ, k)</span>
          </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">Π(n, φ \ α)</span>
          </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">Π(n, φ | m)</span>
          </p></blockquote></div>
<p>
          To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
          of the parameter m</em></span>, or 1 - m, where:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α</span>
          </p></blockquote></div>
<p>
          This implementation uses <span class="emphasis"><em>k</em></span> throughout: this matches
          the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical
          Report on C++ Library Extensions</a>.<br>
        </p>
<p>
          So you should be extra careful when using these functions!
        </p>
</td></tr>
</table></div>
<div class="warning"><table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../doc/src/images/warning.png"></td>
<th align="left">Warning</th>
</tr>
<tr><td align="left" valign="top"><p>
          Boost.Math order of arguments differs from other implementations: <span class="emphasis"><em>k</em></span>
          is always the <span class="bold"><strong>first</strong></span> argument.
        </p></td></tr>
</table></div>
<p>
        A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
        Alpha</a> with Boost.Math (including much higher precision using <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>)
        is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
      </p>
<p>
        When <span class="emphasis"><em>φ</em></span> = <span class="emphasis"><em>π</em></span> / 2, the elliptic integrals
        are called <span class="emphasis"><em>complete</em></span>.
      </p>
<p>
        Complete Elliptic Integral of the First Kind (Legendre form)
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>

        </p></blockquote></div>
<p>
        Complete Elliptic Integral of the Second Kind (Legendre form)
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span>

        </p></blockquote></div>
<p>
        Complete Elliptic Integral of the Third Kind (Legendre form)
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>

        </p></blockquote></div>
<p>
        Legendre also defined a fourth integral /D(φ,k)/ which is a combination of
        the other three:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>

        </p></blockquote></div>
<p>
        Like the other Legendre integrals this comes in both complete and incomplete
        forms.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h2"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals">Carlson
        Elliptic Integrals</a>
      </h5>
<p>
        Carlson [<a class="link" href="ellint_intro.html#ellint_ref_carlson77">Carlson77</a>] [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
        gives an alternative definition of elliptic integral's canonical forms:
      </p>
<p>
        Carlson's Elliptic Integral of the First Kind
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>

        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
        are nonnegative and at most one of them may be zero.
      </p>
<p>
        Carlson's Elliptic Integral of the Second Kind
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>

        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span> are nonnegative, at
        most one of them may be zero, and <span class="emphasis"><em>z</em></span> must be positive.
      </p>
<p>
        Carlson's Elliptic Integral of the Third Kind
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>

        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
        are nonnegative, at most one of them may be zero, and <span class="emphasis"><em>p</em></span>
        must be nonzero.
      </p>
<p>
        Carlson's Degenerate Elliptic Integral
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>

        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>x</em></span> is nonnegative and <span class="emphasis"><em>y</em></span> is
        nonzero.
      </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
          <span class="emphasis"><em>R<sub>C</sub>(x, y) = R<sub>F</sub>(x, y, y)</em></span>
        </p>
<p>
          <span class="emphasis"><em>R<sub>D</sub>(x, y, z) = R<sub>J</sub>(x, y, z, z)</em></span>
        </p>
</td></tr>
</table></div>
<p>
        Carlson's Symmetric Integral
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h3"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.duplication_theorem"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.duplication_theorem">Duplication
        Theorem</a>
      </h5>
<p>
        Carlson proved in [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
        that
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h4"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_s_formulas"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_s_formulas">Carlson's
        Formulas</a>
      </h5>
<p>
        The Legendre form and Carlson form of elliptic integrals are related by equations:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint14.svg"></span>

        </p></blockquote></div>
<p>
        In particular,
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint15.svg"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h5"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals">Miscellaneous
        Elliptic Integrals</a>
      </h5>
<p>
        There are two functions related to the elliptic integrals which otherwise
        defy categorisation, these are the Jacobi Zeta function:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>

        </p></blockquote></div>
<p>
        and the Heuman Lambda function:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>

        </p></blockquote></div>
<p>
        Both of these functions are easily implemented in terms of Carlson's integrals,
        and are provided in this library as <a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">jacobi_zeta</a>
        and <a class="link" href="heuman_lambda.html" title="Heuman Lambda Function">heuman_lambda</a>.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h6"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.numerical_algorithms"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.numerical_algorithms">Numerical
        Algorithms</a>
      </h5>
<p>
        The conventional methods for computing elliptic integrals are Gauss and Landen
        transformations, which converge quadratically and work well for elliptic
        integrals of the first and second kinds. Unfortunately they suffer from loss
        of significant digits for the third kind.
      </p>
<p>
        Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
        [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>], by contrast, provides
        a unified method for all three kinds of elliptic integrals with satisfactory
        precisions.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_intro.h7"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.references"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.references">References</a>
      </h5>
<p>
        Special mention goes to:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          A. M. Legendre, <span class="emphasis"><em>Traité des Fonctions Elliptiques et des Integrales
          Euleriennes</em></span>, Vol. 1. Paris (1825).
        </p></blockquote></div>
<p>
        However the main references are:
      </p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
            <a name="ellint_ref_AS"></a>M. Abramowitz and I. A. Stegun (Eds.) (1964)
            Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
            Tables, National Bureau of Standards Applied Mathematics Series, U.S.
            Government Printing Office, Washington, D.C.
          </li>
<li class="listitem">
            <a href="https://dlmf.nist.gov/19" target="_top">NIST Digital Library of Mathematical
            Functions, Elliptic Integrals, B. C. Carlson</a>
          </li>
<li class="listitem">
            <a name="ellint_ref_carlson79"></a>B.C. Carlson, <span class="emphasis"><em>Computing
            elliptic integrals by duplication</em></span>, Numerische Mathematik,
            vol 33, 1 (1979).
          </li>
<li class="listitem">
            <a name="ellint_ref_carlson77"></a>B.C. Carlson, <span class="emphasis"><em>Elliptic Integrals
            of the First Kind</em></span>, SIAM Journal on Mathematical Analysis,
            vol 8, 231 (1977).
          </li>
<li class="listitem">
            <a name="ellint_ref_carlson78"></a>B.C. Carlson, <span class="emphasis"><em>Short Proofs
            of Three Theorems on Elliptic Integrals</em></span>, SIAM Journal on Mathematical
            Analysis, vol 9, 524 (1978).
          </li>
<li class="listitem">
            <a name="ellint_ref_carlson81"></a>B.C. Carlson and E.M. Notis, <span class="emphasis"><em>ALGORITHM
            577: Algorithms for Incomplete Elliptic Integrals</em></span>, ACM Transactions
            on Mathematical Software, vol 7, 398 (1981).
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em>On computing elliptic integrals and functions</em></span>.
            J. Math. and Phys., 44 (1965), pp. 36-51.
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the second
            kind</em></span>. Math. Comp., 49 (1987), pp. 595-606. (Supplement, ibid.,
            pp. S13-S17.)
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the third kind</em></span>.
            Math. Comp., 51 (1988), pp. 267-280. (Supplement, ibid., pp. S1-S5.)
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: cubic cases</em></span>.
            Math. Comp., 53 (1989), pp. 327-333.
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: one quadratic
            factor</em></span>. Math. Comp., 56 (1991), pp. 267-280.
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: two quadratic
            factors</em></span>. Math. Comp., 59 (1992), pp. 165-180.
          </li>
<li class="listitem">
            B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical
            computation of real or complex elliptic integrals</a></em></span>.
            Numerical Algorithms, Volume 10, Number 1 / March, 1995, p13-26.
          </li>
<li class="listitem">
            B. C. Carlson and John L. Gustafson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9310223" target="_top">Asymptotic
            Approximations for Symmetric Elliptic Integrals</a></em></span>, SIAM
            Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303.
          </li>
</ol></div>
<p>
        The following references, while not directly relevant to our implementation,
        may also be of interest:
      </p>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
            R. Burlisch, <span class="emphasis"><em>Numerical Computation of Elliptic Integrals and
            Elliptic Functions.</em></span> Numerical Mathematik 7, 78-90.
          </li>
<li class="listitem">
            R. Burlisch, <span class="emphasis"><em>An extension of the Bartky Transformation to Incomplete
            Elliptic Integrals of the Third Kind</em></span>. Numerical Mathematik
            13, 266-284.
          </li>
<li class="listitem">
            R. Burlisch, <span class="emphasis"><em>Numerical Computation of Elliptic Integrals and
            Elliptic Functions. III</em></span>. Numerical Mathematik 13, 305-315.
          </li>
<li class="listitem">
            T. Fukushima and H. Ishizaki, <span class="emphasis"><em><a href="http://adsabs.harvard.edu/abs/1994CeMDA..59..237F" target="_top">Numerical
            Computation of Incomplete Elliptic Integrals of a General Form.</a></em></span>
            Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July,
            1994, 237-251.
          </li>
</ol></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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